Schwere, Elektricität und Magnetismus:341

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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Riemann’s Grundgesetz.

T={\frac {1}{2}}m(x'^{2}+y'^{2}+z'^{2})+{\frac {1}{2}}m_{1}(x_{1}'^{2}+y_{1}'^{2}+z_{2}'^{2}),\,

also

{\frac {\partial T}{\partial x'}}=mx',\quad {\frac {\partial T}{\partial x}}=0.

Demnach haben wir jetzt

(3)	$X=m{\frac {d^{2}x}{dt^{2}}}={\frac {d\left({\frac {\partial D}{\partial x'}}\right)}{dt}}-{\frac {\partial D}{\partial x}}+{\frac {\partial S}{\partial x}}.$

Es findet sich aber aus der Formel (II) des vorigen Paragraphen

{\frac {\partial D}{\partial x'}}=2{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r}}\left({\frac {dx}{dt}}-{\frac {x_{1}}{dt}}\right),

${\frac {\partial D}{\partial x}}=-{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial x}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.\,$

Endlich ist

{\frac {\partial S}{\partial x}}={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial x}}.

Setzt man dies ein in Gleichung (3), so erhält man

(4)

X={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial x}}+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {d{\Bigg \lbrace }{\frac {2}{r}}\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right){\Bigg \rbrace }}{dt}}

$+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial x}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.$

Ebenso findet sich

(5)

Y={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial y}}+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {d{\Bigg \lbrace }{\frac {2}{r}}\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right){\Bigg \rbrace }}{dt}}

$+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial y}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.$

(6)

Z={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial z}}+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {d{\Bigg \lbrace }{\frac {2}{r}}\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right){\Bigg \rbrace }}{dt}}

$+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial z}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.$

Ueber bewegte freie Elektricität ist es bis jetzt nicht gelungen, Versuche anzustellen.